AMC 10 · 2025 · #4

Grade 8 algebra
Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation
mean-median-mode-rangesystems-of-equationslinear-equations-two-var convert-to-algebra ↑ Prerequisites: mean-median-mode-rangesystems-of-equations
📏 Medium solution 💡 3 insights
Problem
A student team and a teacher team together have 15 people. If a person named Ash joins the students, their average age goes from 12 to 14; if Ash instead joins the teachers, their average age goes from 55 to 52. Find Ash's age.

Pick an answer.

(A)
28
(B)
29
(C)
30
(D)
32
(E)
33

AMC 10 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

The clues are in words about averages, so Tool #4 (Introduce a Variable) names the unknowns — team sizes S, T and Ash's age A — so every sentence can become an equation. Tool #13 (Convert to Algebra) turns each 'new average' condition into a linear equation using total = average × count. Tool #7 (Identify Subproblems) keeps the student side and the teacher side separate until each has yielded its own equation, after which the two equations plus the total are solved together.

1STEP 1

Name the unknowns

Let S be the number of students, T the teachers, and A Ash's age; the two teams together give S+T=15.

S+T=15
2STEP 2

Turn the students' clue into an equation

The S students total 12S; with Ash there are S+1 people averaging 14, so 12S+A=14(S+1), which gives A=2S+14.

(12S+A)/(S+1)=14 → A=2S+14
3STEP 3

Turn the teachers' clue into an equation

The T teachers total 55T; with Ash there are T+1 people averaging 52, so 55T+A=52(T+1), which gives A=52-3T.

(55T+A)/(T+1)=52 → A=52-3T
4STEP 4

Solve the system for the team sizes

Both equal A, so 2S+14=52-3T, i.e. 2S+3T=38. Substituting S=15-T gives 30+T=38, so T=8 and S=7.

2S+3T=38, S+T=15 → T=8, S=7
5STEP 5

Find Ash's age

Put S=7 into A=2S+14: A=2(7)+14=28. The teachers' equation agrees, 52-3(8)=28, so Ash is 28 — choice (A).

A=2(7)+14=28 = 52-3(8)
Answer
28
An age of 28 sits well above the students (average 12) and well below the teachers (average 55), so it makes sense that Ash pulls the young team's average up and the older team's average down. Both age equations independently return 28, a strong self-check. The near-miss choices 29 through 33 are there to catch a slip such as putting S or T in the denominator instead of S+1 or T+1, which would produce a different system and a wrong age.
💡Key takeaway

Turn each 'new average' clue into 'total = average × count,' then solve the two equations together.

  • Name the unknowns
  • Turn the students' clue into an equation
  • Turn the teachers' clue into an equation
  • Solve the system for the team sizes
  • Find Ash's age

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